Let$^1$
- $U$ and $H$ be separable $\mathbb R$-Hilbert spaces
- $L\in\mathfrak L(H)$ and $B\in\operatorname{HS}(U,H)$
It can be shown that $$\operatorname{tr}LBB^\ast=\operatorname{tr}B^\ast LB<\infty\;.\tag 1$$ Now, let
- $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$
- $U_0:=Q^{1/2}U$
- $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$
- $F:H\to\mathbb R$ be twice Fréchet differentiable and ${\rm D}^2F(x)\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$ denote the second Fréchet derivative of $F$ at $x\in H$
- $\Phi:H\to\operatorname{HS}(U_0,H)$
Let $x\in H$ and chosse $L={\rm D}^2F(x)$ and $B=\Phi(x)Q^{1/2}$. By Riesz' representation theorem, $\exists!\tilde L(x)\in\mathfrak L(H)$ with $$(Ly)z=\langle z,\tilde Ly\rangle_H\;\;\;\text{for all }y,z\in H\tag 2$$ and hence $$\infty>\operatorname{tr}LBB^\ast=\sum_{n\in\mathbb N}\langle Be_n,\tilde LBe_n\rangle_H=\sum_{n\in\mathbb N}(LBe_n)(Be_n)\tag 3\;.$$
Question: Let $$G(x):=\operatorname{tr}\left[{\rm D}^2F(x)\left(\Phi(x)Q^{1/2}\right)\left(\Phi(x)Q^{1/2}\right)^\ast\right]\;\;\;\text{for }x\in H\;.$$ Can we show that $$|G(x)-G(y)|\le c\left\|x-y\right\|_H\;\;\;\text{for all }x,y\in H\tag 4$$ for some $c>0$?
$^1$ Let $\mathfrak L(A,B)$ and $\operatorname{HS}(A,B)$ denote the spaces of bounded linear operators and Hilbert-Schmidt operators, respectively. Moreover, let $\mathfrak L(A):=\mathfrak L(A,A)$.